Final answer:
The distance between the given skew lines is zero since both lines intersect at the origin after solving for points by setting t and s to zero. The explanatory steps provided elucidate how to find the distance in a general case where the lines do not intersect.
Step-by-step explanation:
To find the distance between skew lines, we need to find a vector that is perpendicular to both lines and use it to create a vector between points on each line. The parametric equations given are:
- Line 1: x = 2t, y = 36t, z = 2t
- Line 2: x = 22s, y = 414s, z = -35s
The direction vectors of the lines can be found from the coefficients of t and s respectively:
- Direction of Line 1: ɣ1 = (2, 36, 2)
- Direction of Line 2: ɣ2 = (22, 414, -35)
The vector perpendicular to both is the cross product of the direction vectors:
Perpendicular Vector = ɣ1 x ɣ2
Let's find a point on each line by letting t = 0 for Line 1 and s = 0 for Line 2, we obtain:
- Point on Line 1: P1 = (0, 0, 0)
- Point on Line 2: P2 = (0, 0, 0)
However, because both points are at the origin, the distance is obviously zero. But in a more general case where the points are not the same, we would use the following steps:
- Construct a vector ɣ3 = P2 - P1
- Find the distance d using the dot product between ɣ3 and the Perpendicular Vector
- Finally, divide by the magnitude of the Perpendicular Vector to get the shortest distance d